(Evangelista Torricelli, 1644)

If one rotates a hyperbola around an asymptote, as around an axis, one generates a solid infinite in length in the direction of the axis, which we call an acute hyperbolic solid…It may seem incredible that although this solid has an infinite length, nevertheless none of the cylindrical surfaces we considered has an infinite length but all of them are finite, as will be clear to everyone who is even modestly familiar with the doctrine of Conics.

Torricelli’s Trumpet, or Gabriel’s horn, is a staple of calculus textbooks. It may also be the perfect imaginary musical instrument: an object constituted by its precise mathematical description, yet impossible – due to its infinite length – to realize in material form. This geometrical object was discovered by Galileo Galilei’s student and successor Evangelista Torricelli (also famous for the invention of the barometer). Using Cavalieri’s theory of indivisibles, a predecessor to calculus, Torrelli demonstrated that the object, though infinite in length, has finite volume. Because it contradicted the intuition that an object of infinite length would have infinite volume, the demonstration immediately captured the attention of geometers and philosophers and prompted a reevaluation of the relationship between mathematical and physical reality. Whereas the Pythagorean tradition maintained that geometrical principles governed the material world (“everything is number”), the counterintuitive behavior of infinity suggested a cleft between the ideal truths of mathematics and the realities of physical space.
Torricelli called his geometrical object the hyperbolicum acutum (“acute hyperbolic solid”). The names “Torricelli’s Trumpet” and “Gabriel’s Horn” seem to have come into use much later, perhaps with the modern calculus textbook. The latter name refers to the archangel Gabriel and the trumpet that will announce the Day of Judgment, figuring the physical realization of Torricelli’s geometrical object as an apocalyptic incarnation of the divine. With its mind-bending combination of infinite and finite dimensions, Torricelli’s Trumpet describes an interpenetration of the ideal and the material, and invites us to imagine a music at the intersection of the impossible and the real.
Text: Evangelista Torricelli, Opera Geometrica (Florence, 1644), trans. Paolo Mancosu, Philosophy of Mathematics and Mathematical Practice in the Seventeenth Century (Oxford University Press, 1996)
Images: John Harris Lexicon Technicum, vol 2 (1723); RokerHRO